Physlib.QFT.QED.AnomalyCancellation.Even.Parameterization

For a pure $U(1)$ gauge theory with $2(n+1)$ Weyl fermions, given rational coefficient vectors $g \in \mathbb{Q}^{n+1}$ and $f \in \mathbb{Q}^n$, and a scalar $a \in \mathbb{Q}$, this definition constructs a charge vector $x \in \mathbb{Q}^{2(n+1)}$ that satisfies the linear anomaly cancellation condition (ACC), $\sum_{i=1}^{2n+2} x_i = 0$. The charge vector is defined by the formula: $$x = a \cdot \left( \mathcal{A}(P^!(f), P^!(f), P(g)) \cdot P'(g) - \mathcal{A}(P(g), P(g), P^!(f)) \cdot P^!{}'(f) \right)$$ where $\mathcal{A}(u, v, w) = \sum u_i v_i w_i$ is the symmetric trilinear form associated with the cubic anomaly condition, and $P, P^!, P', P^!{}'$ represent linear embeddings of the parameter spaces into the charge space.

In a pure $U(1)$ gauge theory with $2(n+1)$ Weyl fermions, given rational vectors $g \in \mathbb{Q}^{n+1}$ and $f \in \mathbb{Q}^n$ and a rational scalar $a$, the charge vector $x \in \mathbb{Q}^{2n+2}$ obtained from the linear parameterization is given by the formula: $$x = a \cdot \left( \mathcal{A}(P^!(f), P^!(f), P(g)) \cdot P(g) - \mathcal{A}(P(g), P(g), P^!(f)) \cdot P^!(f) \right)$$ where $\mathcal{A}(u, v, w) = \sum_i u_i v_i w_i$ is the symmetric trilinear form associated with the cubic anomaly cancellation condition, and $P$ and $P^!$ are linear embeddings from the parameter spaces into the space of charges.

In a pure $U(1)$ gauge theory with $2n+2$ Weyl fermions, let $g \in \mathbb{Q}^{n+1}$ and $f \in \mathbb{Q}^n$ be rational vectors and $a \in \mathbb{Q}$ be a rational scalar. Let $x \in \mathbb{Q}^{2n+2}$ be the charge vector constructed via the linear parameterization: $$x = a \cdot \left( \mathcal{A}(P^!(f), P^!(f), P(g)) \cdot P(g) - \mathcal{A}(P(g), P(g), P^!(f)) \cdot P^!(f) \right)$$ where $\mathcal{A}(u, v, w) = \sum_{i=1}^{2n+2} u_i v_i w_i$ is the symmetric trilinear form associated with the cubic anomaly, and $P$ and $P^!$ are linear embeddings of the parameter spaces into the charge space. Then the vector $x$ satisfies the cubic anomaly cancellation condition (ACC): $$\sum_{i=1}^{2n+2} x_i^3 = 0$$

For a pure $U(1)$ gauge theory with an even number of fermions $2(n+1)$, this definition constructs a charge vector $x \in \mathbb{Q}^{2n+2}$ that satisfies both the linear and cubic anomaly cancellation conditions (ACC), namely $\sum_{i=1}^{2n+2} x_i = 0$ and $\sum_{i=1}^{2n+2} x_i^3 = 0$. Given rational parameter vectors $g \in \mathbb{Q}^{n+1}$ and $f \in \mathbb{Q}^n$ and a scaling factor $a \in \mathbb{Q}$, the solution is defined as: $$x = a \cdot \left( \mathcal{A}(P^!(f), P^!(f), P(g)) \cdot P(g) - \mathcal{A}(P(g), P(g), P^!(f)) \cdot P^!(f) \right)$$ where $\mathcal{A}(u, v, w) = \sum_{i=1}^{2n+2} u_i v_i w_i$ is the symmetric trilinear form associated with the cubic anomaly, and $P$ and $P^!$ are the linear embeddings of the parameter spaces into the charge space.

Consider a pure $U(1)$ gauge theory with $2n+2$ Weyl fermions, where the space of charges is $\mathbb{Q}^{2n+2}$. Let $S$ be a vector of rational charges that satisfies the anomaly cancellation conditions (ACC), which include the cubic condition $\sum_{i=1}^{2n+2} S_i^3 = 0$. Suppose $S$ can be decomposed as $S = P(g) + P^!(f)$, where $g \in \mathbb{Q}^{n+1}$ and $f \in \mathbb{Q}^n$ are vectors of parameters, and $P$ and $P^!$ are the linear maps associated with the parameterization of the even case. Then, the symmetric trilinear form $\mathcal{A}(x, y, z) = \sum_{i=1}^{2n+2} x_i y_i z_i$ satisfies the following identity: $$\mathcal{A}(P(g), P(g), P^!(f)) = -\mathcal{A}(P^!(f), P^!(f), P(g)).$$

Consider a pure $U(1)$ gauge theory with an even number of fermions $2(n+1)$, where solutions to the anomaly cancellation conditions are vectors of rational charges $S \in \mathbb{Q}^{2(n+1)}$. The solution $S$ is said to be in the **generic case** if, for every pair of parameter vectors $g \in \mathbb{Q}^{n+1}$ and $f \in \mathbb{Q}^n$ such that $S$ is decomposed as $S = P(g) + P!(f)$, the symmetric trilinear form $\mathcal{A}(x, y, z) = \sum_{i=1}^{2(n+1)} x_i y_i z_i$ satisfies: $$\mathcal{A}(P(g), P(g), P!(f)) \neq 0$$ where $P$ and $P!$ are the specific mappings used in the parameterization of the even-dimensional charge space.

Consider a pure $U(1)$ gauge theory with $2(n+1)$ fermions, where a solution $S$ to the anomaly cancellation conditions is a vector of rational charges $S \in \mathbb{Q}^{2n+2}$ satisfying $\sum_{i=1}^{2n+2} S_i = 0$ and $\sum_{i=1}^{2n+2} S_i^3 = 0$. If there exist parameter vectors $g \in \mathbb{Q}^{n+1}$ and $f \in \mathbb{Q}^n$ such that $S$ is decomposed as $S = P(g) + P^!(f)$ and the symmetric trilinear form $\mathcal{A}(x, y, z) = \sum_{i=1}^{2n+2} x_i y_i z_i$ satisfies $$\mathcal{A}(P(g), P(g), P^!(f)) \neq 0,$$ then $S$ is classified as being in the **generic case**.

Let $S$ be a solution to the anomaly cancellation conditions (ACC) for a pure $U(1)$ gauge theory with $2(n+1)$ Weyl fermions. The solution $S$ is said to be in the **special case** if, for every possible parameterization of $S$ as $S = P g + P! f$ (where $g \in \mathbb{Q}^{n+1}$ and $f \in \mathbb{Q}^n$), the symmetric trilinear form $\mathcal{A}$ satisfies: $$\mathcal{A}(P g, P g, P! f) = 0$$ where $\mathcal{A}(x, y, z) = \sum_{i=1}^{2n+2} x_i y_i z_i$ is the cubic anomaly form.

Let $S$ be a solution to the anomaly cancellation conditions for a pure $U(1)$ gauge theory with $2(n+1)$ Weyl fermions, where the solution is represented by a vector of rational charges $x \in \mathbb{Q}^{2n+2}$ satisfying $\sum x_i = 0$ and $\sum x_i^3 = 0$. If there exist mappings $g : \text{Fin}(n+1) \to \mathbb{Q}$ and $f : \text{Fin}(n) \to \mathbb{Q}$ such that the charge vector of $S$ is given by $S = P g + P! f$ and the symmetric trilinear form satisfies $\mathcal{A}(P g, P g, P! f) = 0$ (where $\mathcal{A}(x, y, z) = \sum_{i=1}^{2n+2} x_i y_i z_i$), then $S$ is in the special case.

For any solution $S$ to the anomaly cancellation conditions (ACC) of a pure $U(1)$ gauge theory with an even number of fermions $2(n+1)$, where $S$ is a vector of rational charges $(x_1, \dots, x_{2n+2}) \in \mathbb{Q}^{2n+2}$ satisfying $\sum_{i=1}^{2n+2} x_i = 0$ and $\sum_{i=1}^{2n+2} x_i^3 = 0$, $S$ is classified as either being in the **generic case** or the **special case**. A solution $S$ is in the **generic case** if there exist parameter vectors $g \in \mathbb{Q}^{n+1}$ and $f \in \mathbb{Q}^n$ such that $S$ is decomposed as $S = P(g) + P^!(f)$ and the symmetric trilinear form $\mathcal{A}(x, y, z) = \sum_{i=1}^{2n+2} x_i y_i z_i$ satisfies $\mathcal{A}(P(g), P(g), P^!(f)) \neq 0$. Otherwise, $S$ is in the **special case**.

Consider a pure $U(1)$ gauge theory with $2(n+1)$ Weyl fermions. Let $S = (x_1, \dots, x_{2n+2}) \in \mathbb{Q}^{2n+2}$ be a vector of rational charges that satisfies the anomaly cancellation conditions (ACC), $\sum_{i=1}^{2n+2} x_i = 0$ and $\sum_{i=1}^{2n+2} x_i^3 = 0$. If $S$ is in the **generic case**, then there exist rational parameter vectors $g \in \mathbb{Q}^{n+1}$ and $f \in \mathbb{Q}^n$, and a rational scaling factor $a \in \mathbb{Q}$, such that $S$ can be expressed by the parameterization: $$S = a \cdot \left( \mathcal{A}(P^!(f), P^!(f), P(g)) \cdot P(g) - \mathcal{A}(P(g), P(g), P^!(f)) \cdot P^!(f) \right)$$ where $\mathcal{A}(u, v, w) = \sum_{i=1}^{2n+2} u_i v_i w_i$ is the symmetric trilinear form associated with the cubic anomaly, and $P$ and $P^!$ are the linear embeddings of the parameter spaces into the charge space.

For a pure $U(1)$ gauge theory with $2n+2$ Weyl fermions, let $S \in \mathbb{Q}^{2n+2}$ be a vector of charges that satisfies the anomaly cancellation conditions (ACC), specifically $\sum_{i=1}^{2n+2} S_i = 0$ and $\sum_{i=1}^{2n+2} S_i^3 = 0$. If $S$ falls under the **special case**, meaning that for every decomposition of $S$ into the form $P g + P^! f$ (where $g \in \mathbb{Q}^{n+1}$ and $f \in \mathbb{Q}^n$), the symmetric trilinear form $\mathcal{A}(x, y, z) = \sum x_i y_i z_i$ satisfies $\mathcal{A}(P g, P g, P^! f) = 0$, then $S$ is a **line in the cubic** (i.e., it belongs to a specific linear subspace of the cubic anomaly surface).

Let $S = (x_1, \dots, x_{2n+2}) \in \mathbb{Q}^{2n+2}$ be a vector of charges satisfying the pure $U(1)$ anomaly cancellation conditions $\sum_{i=1}^{2n+2} x_i = 0$ and $\sum_{i=1}^{2n+2} x_i^3 = 0$. If for every permutation $M \in S_{2n+2}$, the permuted solution $M(S)$ satisfies the **special case** condition (meaning that for every decomposition $M(S) = P g + P^! f$, the cubic form satisfies $\mathcal{A}(P g, P g, P^! f) = 0$), then $S$ is a **line in the cubic** up to permutation (denoted `LineInCubicPerm`).

Let $S = (x_1, \dots, x_{2n+4}) \in \mathbb{Q}^{2n+4}$ be a vector of charges satisfying the anomaly cancellation conditions for a pure $U(1)$ gauge theory with $2n+4$ Weyl fermions (specifically, $\sum x_i = 0$ and $\sum x_i^3 = 0$). If for every permutation $M$ in the symmetric group $S_{2n+4}$, the permuted solution $M(S)$ satisfies the **special case** condition—meaning that for every decomposition $M(S) = P g + P^! f$, the cubic anomaly form satisfies $\mathcal{A}(P g, P g, P^! f) = 0$—then there exists a permutation $M \in S_{2n+4}$ such that the resulting charge vector $M(S)$ lies in the $\mathbb{Q}$-linear subspace spanned by the range of the basis vectors.